B How Does Separating the Foci Affect the Shape of a Spheroid?

[DaveC426913]

A story I wrote depends on some geometry, and I want to get it straight.

Assumptions:
1] The 2D math that applies to circles and ellipses is analagous to the 3D math for spheres and ellipsoids in the ways relevant to the rest of my post.
2] An ellipse is a circle whose two foci are co-incident. Thus, a sphere is an ellipsoid whose two foci are co-incient.

What I want to figure out is what happens to the surface of an ellipse/ellipsoid when the two co-incident foci are moved apart.

Say I have a sphere of unit radius. I move its two foci two units apart. What happens to the surface?
What is the length of its semi-minor axis?
What is the length of its semi-major axis?

And, more specifically, what happens to the distance from focus-to-surface near the "ends"?
Does the distance from focus to surface decrease as the ellipsoid "stretches" thinner? (i.e. is the focus now closer to the surface than one unit?)[ EDIT ] The correct term for my shape is a prolate spheroid - a cigar shape**. Equatorial radii a and b are the same. Polar radius c is longer.

** by magical coincidence, I am writing this while smoking a cigar.

 

[mfb]

Say I have a sphere of unit radius. I move its two foci two units apart. What happens to the surface?
What is the length of its semi-minor axis?
What is the length of its semi-major axis?

Same as in the two-dimensional case. The ellipsoid you get in that way is the ellipse rotated around the symmetry axis. Note that not all ellipsoids can be produced that way - you are limited to those with one large axis and two identical smaller axes.
I wonder what shape you get if you require the sum of distances to three (instead of two) points to be constant. The two-dimensional shape can be egg-like. Another egg.

 

[fresh_42]

I wonder what shape you get if you require the sum of distances to three (instead of two) points to be constant. The two-dimensional shape can be egg-like.

Some strange eggs, I'd say:

http://arxiv.org/pdf/math/0702005v1.pdf

http://jwilson.coe.uga.edu/emt725/Tri.Locus/FourPt/FourPt.html

Sorry, for getting off topic.

 

[DaveC426913]

Same as in the two-dimensional case. The ellipsoid you get in that way is the ellipse rotated around the symmetry axis.

Right. Which is what?

(The only reason I mentioned the 2D case is so that, when I set up some details in my story, I can sketch the simpler 2D geometry. I don't have to wory about whether 3D geometry has different behavior).

 

[DaveC426913]

The end-result of my question is this: as the two foci separate, does the focus-to-surface distance (what was a moment ago, the radius) at the pole decrease to less than one unit?

 

[fresh_42]

** by magical coincidence, I am writing this while smoking a cigar.

... as long as the NFL is off season ...

The end-result of my question is this: as the two foci separate, does the focus-to-surface distance (what was a moment ago, the radius) at the pole decrease to less than one unit?

I have a question. Why isn't the equation for the eccentricity $e^2 = 1 - \frac{a^2}{c^2}$ from Wiki (https://en.wikipedia.org/wiki/Spheroid) not the answer because the planar foci are $(0,±e)$?

 

[DaveC426913]

Ah. OK. Once I drew it out, it was pretty simple. Should have started with that.

So, the distance I was looking for was BP = .5

 

[DaveC426913]

Shoot. So THAT's a problem. The above ellipse assumes that it is the radii that remain constant. It can not grow larger than a major axis of 2. At that point, it degenerates to a line. That's useless to me.

I'm going to need to redo it using 3D volume as a constant.

So, a sphere of one unit radius has a volume of 4/3π. I need to find the major and minor axes of a prolate elliptoid with a volume of 4/3π whose foci are one unit apart.Volume of sphere s = 4/3π*r³
Volume of prolate elliptoid e = 4/3π*a²c

e=s
4/3π*r³ = 4/3π*a²c
r=a²c
a²c = 1

Hm. Missing something here...
I need to specify a as a ratio of c.

How does the focal separation relate to the major and minor axes?
More specifically, how can I specify a and c so that the foci are one unit apart?

 

[DaveC426913]

I have a question. Why isn't the equation for the eccentricity $e^2 = 1 - \frac{a^2}{c^2}$ from Wiki (https://en.wikipedia.org/wiki/Spheroid) not the answer

No idea, sorry.